Being very limited in my knowledge of the aviation world, especially outside of the U.S. I have questions about various Aviation Administrations around the world. Are there big differences in the rules and regs between the FAA and others around the world? Are there any rules that non US carriers have to abide by while in the US that they may not have to follow in their home country? Is one country's FAA more strict than others?

"But this airplane has 4 engines, it's an entirely different kind of flying! Altogether"

ICAO is the United Nations body that establishes overall international aviation policies, subject to ratification by each UN member country. Most countries have many exceptions to the ICAO Annexes.

Also, ICAO doesn't make rules per se, but rather "recommendations", to put it one way, and contracting countries are encouraged to abide by them. That doesn't stop a bunch of countries doing whatever the heck they want and making absurd regulations

Very much so, in different ways.
Example.
I used to fly an L1011 for one overseas airline that followed FAA flight deck crew duty time limitations, which were 14 hours duty time (single crew), 21 hours (augmented crew) 24 hours (double crew).
Now, I fly an L1011 for a different company, and their CAA approved crew duty time limits are more relaxed, IE: 16 hours duty (single crew), 23 hours (augmented crew), 27 hours (double crew).